I agree, it is always a good idea to make sure your computer is clean before you start trying to solve other problems. In view of your remarks about having a lot of stuff on here a good starting point would be to download and install Piriform's Ccleaner, an excellent little utility for removing temporary files. It's a free download and you can get it from BC here -
Just one word of advice. Ccleaner contains a registry 'cleaner' as one of its sections. Do NOT use this part of it.
As you have been though the getting rid of malware process, can I assume that you have up to date antivirus on your computer ? If you do, run a full system scan and see if that shows up any problems. If it does, let the AV deal with them.
As far as back-ups go, what matters is you back-up your own work - photos, music, documents, etc.. This is the stuff that you cannot replace if a hard drive decides to die on you. Since I keep my own work in a folder called 'Workfile', my photos in a folder called 'Photos' and my music in a folder called 'Music' as opposed to using the Windows defaults this makes doing back-ups fairly simple and I just use Windows Explorer and 'drag and drop' to do back-ups.
Once you have backed up stuff to your external drive you can then delete it from you internal drive to create space on that. This then leaves you with only one copy - on your external drive - which for safety should be backed up to another drive. External hard drives can fail and only one copy of anything in existence is not a back up..
Run Ccleaner and see how much junk it gets rid of, do an AV scan, and let us know how you get on. Once your computer itself is running better it will then be time to start a new topic in the 'Networking' section of BC so the people there can give advice on your connection problems.